Page 17 - Fall 2008

cation of scattering. When the masses are infinite, then the motion of a length of string between two masses can have no effect on any other section of string—thus the infinite mass- es divide the string into “isolated local oscillators.” Each local oscillator consists of a string of length a, clamped at both ends. The normal modes of each local oscillator consist of fit- ting an integer number of half-wavelengths into the length a, and the natural frequencies form a harmonic sequence with fν = νc/2a. The frequencies with ν equal to 0, 1 and 2 are the ones on the right in Fig. 6, and the sequence continues upward to infinity. When isolated with the equivalent of infi- nite masses, the local oscillator is said to have “sharp” natural frequencies. If the scatterer mass is reduced from infinity to a finite value (as illustrated by the vertical dashed line in Fig. 6) then a mass might be moved by the motion of one local oscil- lator string, and affect the next local oscillator. In other words, the case of finite scatterer mass means that the local oscillators become coupled together. The result is analogous to what happens when two or more simple mass-spring oscil- lators are coupled together. A number of new normal modes and natural frequencies arise. Thus the result of coupling the local oscillators is that the sharp frequencies of an isolated oscillator broaden out into bands, and band structure appears (as seen by moving from the far right toward the left in Fig. 6). In solid state physics, this picture of the evolution of frequencies from the sharp values of isolated local oscilla- tors into bands is called the “tight binding” model.4 In an iso- lated atom, an electron is tightly bound to an ion core, and because of quantum mechanics, the electron has discrete (sharp) energy levels. But when many atoms are put together at periodic lattice sites in a crystal, the electrons may interact with neighboring atoms, and the sharp atomic energy levels broaden out into bands, resulting in the electronic band structure in solid state physics. It should be noted that nearly all modern electronic devices owe their existence to the band structure of semiconductors.
Mode shapes and wave propagation
As discussed above, the symbols along the vertical dashed line in Fig. 6 give the frequencies for the modes of a string that contains periodic scatterers of finite mass m. What about the nature of the modes themselves? When the scatter- er mass was zero, the modes could readily be drawn as in Fig. 5a. When the scatterer mass was infinite, the modes could readily be drawn as in Fig. 5b. However, when the scatterer mass is finite, determining the modes is not at all easy. To draw the modes for the string with finite masses, equivalent to the modes “a” through “q” in Figs. 5 and 6, a computer pro- gram was used with the results shown in Fig. 7. The vertical position of the dashed line for the different modes in the fig- ure is roughly proportional to the frequency of the modes. The gaps between modes “h” and “i” and between modes “p” and “q” are apparent. Some modes, such as pairs “b” and “c,” “d” and “e,” etc., remain degenerate. However, for the plain string the pair of modes “h” and “i” and the pair “p” and “q” were degenerate, but now the presence of finite scatterers has “split” these degeneracies, resulting in the gaps.
For a system with finite mass scatterers, there is an
Fig. 7. The modes of vibration for a string with scatterers of finite mass. The verti- cal position of the dashed line for the different modes is roughly proportional to the frequency of the mode. Some modes remain degenerate. The gaps between modes “h” and “i” and between modes “p” and “q” are apparent.
important type of diagram, called a band structure diagram (Fig. 8) that is used to display graphically some of the infor- mation indicated by the symbols along the vertical dashed line in Fig. 6 and by the features of Fig. 7. More information concerning the band structure diagram, mode modulus, dis- persion, and Bloch waves may be found in the Appendix.
An important feature of wave propagation in periodic media is that there can be no wave propagation at any fre- quency that falls in a stop band or gap. For example, there can be no propagating wave at a frequency that lies between “h” and “i” and between “p” and “q” in Fig. 7. However, when a wave at a gap frequency, traveling in a region free of scatter- ers, encounters a periodic array of scatterers, the wave motion is not stopped abruptly. The wave enters the region of scatterers as an “evanescent” wave7 that decays exponentially. Because there is no energy transmission for an evanescent wave, the energy of the incident wave is completely reflected.
New research with waves in arrays of scatterers
Interest in wave propagation in periodic systems dates
back to the 1880’s. Lord Rayleigh published an article on
8
waves in periodically stratified media in 1887. Later, the sub-
ject intensified considerably with the application of coherent electromagnetic waves (microwaves and light from lasers) in
16 Acoustics Today, October 2008